## Lecture Summaries for COMP
566A
Autumn 2005

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**Text**

Linear Programming, V.
Chvatal

Sept 1 No class

Sept 6: Introduction to Mathematical Programming, and a quick outline
of the history of Linear Programming. We
discussed the fundamental contributions of George
Dantzig, and Leonid
Khachiyan, both of whom passed away this year.

Sept 8: How to solve LPs in standard form. Read ch 2 of the text. Make up and solve one
complete example by hand !

Sept 13: A formal statement of the simplex method. Dantzig and
Bland's rule. Cycling and degeneracy. A geometric view of the simplex
method. Read: text ch. 3 except pp 34-38.

Sep 15: Another simple and self contained way of getting a feasible
solution to a system of inequalities was described. We proved
finiteness
and Farkas' lemma. Read: notes.

Sept 20: Certificates for the 3 termination conditions. Unbounded
solutions. Duality theory. Proof of the weak duality theorems.
Read:
Ch 5 up to p. 62 except proof of thm 5.1.

Sep 22: Proof of Strong duality theorem. Interpretation of dual
variables. Read: Proof of Thm 5.1 and pp. 65-68.

Sep 27: Complementary slackness conditions. Formulation of 2-layer
graph layout as an integer program. Read: pp
62-68 and for graph layout the paper
of Junger et al.

Sep 29. The revised simplex method. Read text: Ch 7 to p. 105.

Oct 4. Worst case of the simplex method, Klee-Minty examples.
Interpretation of dual and complementary slackness for the diet
problem. Read: Ch. 4.

Oct 6: Matrix factorization. Revised simplex methos and complementary
slackness. Read: rest of Ch. 7.

The first class test will be based on
the above material.

Oct. 11: Review problems

Oct. 13: First class test.

Oct. 18: Guest lecturer: Antoine Deza. "Combinatorial
Optimization: Problems and Algorithms
"

Oct. 20: Dual simplex method. Read Chvatal 148-156.

Oct. 25: Integer Programming via cutting planes. Read: Class handout.

Oct. 27: Network Simplex method. Read Chvatal ch. 19 up to p. 307.
Assignment 3 posted.

Nov. 1: Matrix games. Zero sum games and the min-max theorem. Read
text, pp228-233.

Nov. 3. Non-zero sum games and Nash equilibria. Read: Notes
here.

Second class test contains all material up to and including Nov. 3.

Emphasis on material since first test.

Nov. 8,10 Primal-dual interior point method. Read: Class notes.

Nov. 15 Wrap up.

Nov 17: Second class test.

Nov 22,24,29 Class
presentations. Attendance
compulsory!

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